A The information about the curvature on the rest

What happens with the information about the curvature of the trajectory of a body when the body stops? We know that to assess the curvature of a path we must calculate the value of a fraction of which the denominator is speed module (cubed). But if speed module is canceled, this fraction can not be assessed. So, it preserves information about the curvature in that interactions that stop the body?

Thanks for the reply! But, about the curvature of the trajectory, you have an idea? You know what is the curvature? You know how to calculate it? If yes, then what value does it for a body with zero speed?

I was addressed those who already know the formula. I did not open this topic to make education here. The formula can be found, for example, in the Wikipedia. $$\kappa=\frac{|\gamma'\times\gamma''|}{|\gamma'|^3}$$

Thanks for the reply! But, about the curvature of the trajectory, you have an idea? You know what is the curvature? You know how to calculate it? If yes, then what value does it for a body with zero speed?

To ask about curvature you should first see to what "object" this curvature applies to.
So what do you think is the trajectory of a body at rest?

I was addressed those who already know the formula. I did not open this topic to make education here. The formula can be found, for example, in the Wikipedia. $$\kappa=\frac{|\gamma'\times\gamma''|}{|\gamma'|^3}$$

Take the trajectory in given by the parameterization:

$$x(t)=t^3$$
$$y(t)=0$$
$$z(t)==0$$

Take the trajectory given by the parameterization:

$$x(t)=t$$
$$y(t) = 0$$
$$z(t) = 0$$

Do these two define the same path? Does this path include the point (0,0,0)? What is the curvature of the path at (0,0,0)?

Staff: Mentor

Suppose, instead of the projectile coming to rest (as reckoned by one observer), its motion is reckoned by another observer who is traveling in an inertial frame of reference at constant velocity relative to the first observer. According to this observer, once the projectile hits the ground, it does not come to rest. Instead, it continues in the same direction at constant velocity tangent to the ground. So, according to this observer, the curvature of the projectile path has undergone a discontinuous change. How much of the previous curvature is preserved by the straight line path that the projectile is then experiencing after making contact with the ground?

Here, for instance is a coherent version of what you may have meant to ask.

Can we determine the curvature at a particular point on the path traced out by a point particle if we are given only information about the position, velocity and acceleration of the particle at a time when it was at rest at that point?

Edit to add...

Another coherent question would be to ask whether it might be possible to define the curvature of such a path at such a point in some other way. (i.e. using some other parameterization).